Finite Group Databasebeta

← #24919230

Subgroups — #24919230

Order 10624.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128513C211yesmaximal, characteristic, complemented
2#19128508C211yesmaximal, characteristic, complemented
3#19128506C211yesmaximal, characteristic, complemented
4#19128503C211yesmaximal, characteristic, complemented
5#19128090C211yesmaximal, characteristic, complemented
6#19127972C211yesmaximal, characteristic, complemented
7#19127971C211yesmaximal, characteristic, complemented
8#19127965C211yesmaximal, characteristic, complemented
9#19127964C211yesmaximal, characteristic, complemented
10#19127880C211yesmaximal, characteristic, complemented
11#19127875C211yesmaximal, characteristic, complemented
12#19127870C211yesmaximal, characteristic, complemented
13#19127870C211yesmaximal, characteristic, complemented
14#19127824C211yesmaximal, characteristic, complemented
15#444×C83C211yesmaximal, characteristic, complemented
16#14581839C2211yescharacteristic, complemented
17#14581838C2211yescharacteristic
18#14581838C2211yescharacteristic, complemented
19#14581838C2211yescharacteristic, complemented
20#14581825—21no—
21#14581825C2211yescharacteristic, complemented
22#14581824—21no—
23#14581822C2211yescharacteristic, complemented
24#14581820—21no—
25#14581820—21no—
26#14581820—21no—
27#14581820C2211yescharacteristic
28#14581820C2211yescharacteristic, complemented
29#14581820C2211yescharacteristic, complemented
30#14581819—21no—
31#14581819—21no—
32#14581817C2211yescharacteristic
33#14581817C2211yescharacteristic, complemented
34#14581817C2211yescharacteristic, complemented
35#14581816—21no—
36#14581815—21no—
37#14581814—21no—
38#14581814—21no—
39#14581814C2211yescharacteristic
40#14581813—21no—
41C2×#13050579—21no—
42C2×#13050579C2211yescharacteristic, complemented
43C2×#13050579C2211yescharacteristic, complemented
44SD16×D83—21no—
45C2×D664—21no—
46C2×D664C2211yescharacteristic, complemented
47C2×D664C2211yescharacteristic, complemented
48D8×D83—21no—
49#14581798C2211yescharacteristic, complemented
50#14581798C2211yescharacteristic, complemented
51#14581789C2211yescharacteristic, complemented
52#14581780C2211yescharacteristic, complemented
53#14581728C2211yescharacteristic, complemented
54#14581720C2211yescharacteristic, complemented
55#14581720C2211yescharacteristic, complemented
56#14581710C2211yescharacteristic, complemented
57#14581708C2211yescharacteristic, complemented
58#14581708C2211yescharacteristic, complemented
59#14581708C2211yescharacteristic, complemented
60#135×C83C2211yescharacteristic
61#131×C83C2211yescharacteristic, complemented
62#131×C83C2211yescharacteristic, complemented
63#108×C83C2211yescharacteristic, complemented
64#104×C83C2211yescharacteristic, complemented
65#104×C83C2211yescharacteristic, complemented
66#97×C83C2211yescharacteristic, complemented
67#13050591—21no—
68#13050591—21no—
69#13050591—21no—
70#13050590—21no—
71#13050590—21no—
72#13050590—21no—
73#13050590—21no—
74#13050590C2311yescharacteristic
75#13050590C2311yescharacteristic
76#13050589—21no—
77#13050589—41no—
78C2×Dic166—21no—
79C2×Dic166C2311yescharacteristic
80Q8×D83—21no—
81C2×D332—21no—
82C2×D332—21no—
83C2×D332—21no—
84C2×D332—41no—
85C2×D332C2311yescharacteristic
86C2×D332C2311yescharacteristic
87C2×D332C2311yescharacteristic
88D4×D83—21no—
89D4×D83—21no—
90D4×D83—21no—
91D4×D83—41no—
92D4×D83—41no—
93D4×D83—41no—
94Dic332—21no—
95Dic332D411yescharacteristic, complemented
96#13050580—21no—
97#13050579—21no—
98#13050579—21no—
99#13050579—21no—
100#13050579—21no—
101#13050579—21no—
102#13050579—21no—
103#13050579—21no—
104#13050579—21no—
105#13050579D411yescharacteristic, complemented
106#13050579D411yescharacteristic, complemented
107#13050578—21no—
108#13050577—21no—
109#13050577—21no—
110#13050577—21no—
111D664—21no—
112D664—21no—
113D664—21no—
114D664—21no—
115D664—21no—
116D664—21no—
117D664—21no—
118D664D411yescharacteristic, complemented
119#13050575—21no—
120#13050575—21no—
121#13050575—21no—
122#13050574C2311yescharacteristic
123#13050574C2311yescharacteristic
124#13050573—21no—
125#13050573—21no—
126C8×D83—21no—
127C8×D83—21no—
128(C2×D83)⋊C4—41no—
129#41×C83C2311yescharacteristic
130#41×C83C2311yescharacteristic
131Q16×C83D411yescharacteristic, complemented
132SD16×C83D411yescharacteristic, complemented
133SD16×C83D411yescharacteristic, complemented
134D8×C83D411yescharacteristic, complemented
135M16×C83—21no—
136M16×C83—21no—
137M16×C83C2311yescharacteristic
138M16×C83C2311yescharacteristic
139C2×C8×C83—21no—
140C2×C8×C83—21no—
141C2×C8×C83C2311yescharacteristic
142C2×C8×C83C2311yescharacteristic
143C42×C83C2311yescharacteristic
144C22×D83—21no—
145C22×D83—21no—
146C22×D83—21no—
147C22×D83—41no—
148C22×D83—41no—
149C22×D83—41no—
150Q8×C83C2×D411yescharacteristic
151Q8×C83C2×D411yescharacteristic
152D4×C83—21no—
153D4×C83—21no—
154D4×C83C2×D411yescharacteristic
155D4×C83C2×D411yescharacteristic
156C2×C4×C83—21no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83C2411yescharacteristic
160C22⋊D83—21no—
161C22⋊D83—21no—
162C22⋊D83—41no—
163C22⋊D83—41no—
164C22⋊D83—41no—
165C22⋊D83—41no—
166C2×Dic83—21no—
167C2×Dic83—41no—
168D332—21no—
169D332—21no—
170D332—21no—
171D332—21no—
172D332—21no—
173D332—21no—
174D332—41no—
175D332—41no—
176D332—41no—
177D332—82no—
178D332C2×D411yescharacteristic
179D332C2×D411yescharacteristic
180C4×D83—21no—
181C4×D83—21no—
182C4×D83—41no—
183C4×D83—41no—
184C4×D83—41no—
185C4×D83—41no—
186Dic166—21no—
187Dic166—21no—
188Dic166—41no—
189Dic166C2×D411yescharacteristic
190Dic166C2×D411yescharacteristic
191C8×C83—21no—
192C8×C83—21no—
193C8×C83C2×D411yescharacteristic
194C8×C83C2×D411yescharacteristic
195C8×C83C2×D411yescharacteristic
196C8×C83C2×D411yescharacteristic
197C83⋊C8—21no—
198C83⋊C8—21no—
199C22×C83—21no—
200C22×C83—21no—
201C22×C83H(5,𝔽2)11yescharacteristic
202C2×D83—21no—
203C2×D83—21no—
204C2×D83—41no—
205C2×D83—41no—
206C2×D83—41no—
207C2×D83—81no—
208C2×D83—81no—
209C2×D83—81no—
210C2×D83—161no—
211C2×D83—161no—
212C2×D83—161no—
213C4×C83—21no—
214C4×C83—21no—
215C4×C83—41no—
216C4×C83C22×D411yescharacteristic
217C4×C83C22×D411yescharacteristic
218Dic83—21no—
219Dic83—21no—
220Dic83—41no—
221C2×C83—21no—
222C2×C83—41no—
223C2×C83—41no—
224C2×C83D4211yescharacteristic
225D83—41no—
226D83—41no—
227D83—81no—
228D83—81no—
229D83—81no—
230#3288—831nomaximal, Sylow
231C83#328811yescharacteristic, complemented, Sylow
232#577—831no—
233#577—831no—
234#576—831no—
235#576—831no—
236#496—831no—
237#472—831no—
238#472—831no—
239#471—831no—
240#471—831no—
241#455—831no—
242#454—831no—
243#453—831no—
244#453—831no—
245#444—831no—
246#444D8311yescharacteristic, complemented
2472−1+4—831no—
248H(5,𝔽2)—831no—
249H(5,𝔽2)—831no—
250H(5,𝔽2)—831no—
251#137—831no—
252#137—831no—
253#137—1661no—
254#137—1661no—
255AGL(1,ℤ/8)—831no—
256AGL(1,ℤ/8)—831no—
257AGL(1,ℤ/8)—831no—
258AGL(1,ℤ/8)—831no—
259AGL(1,ℤ/8)—831no—
260AGL(1,ℤ/8)—831no—
261AGL(1,ℤ/8)—1661no—
262AGL(1,ℤ/8)—1661no—
263AGL(1,ℤ/8)—1661no—
264AGL(1,ℤ/8)—1661no—
265AGL(1,ℤ/8)—1661no—
266AGL(1,ℤ/8)—1661no—
267#135—831no—
268#135—1661no—
269#135—1661no—
270#135—1661no—
271#135—1661no—
272#135C2×D8311yescharacteristic
273C2×SD16—831no—
274C2×SD16—831no—
275C2×SD16—1661no—
276C2×SD16—1661no—
277C2×D8—831no—
278C2×D8—831no—
279C2×D8—1661no—
280C2×D8—1661no—
281#131—831no—
282#131—831no—
283#131C2×D8311yescharacteristic, complemented
284#131C2×D8311yescharacteristic, complemented
285#127—831no—
286#124—831no—
287#108—831no—
288#108C2×D8311yescharacteristic, complemented
289#104—831no—
290#104—831no—
291#104C2×D8311yescharacteristic, complemented
292#104C2×D8311yescharacteristic, complemented
293#101—831no—
294#100—831no—
295#100—831no—
296#100—831no—
297#97C2×D8311yescharacteristic, complemented
298#41—3321no—
299#41—831no—
300#41—831no—
301#41—1661no—
302#41—1661no—
303#41—1661no—
304#41—1661no—
305#41—1661no—
306#41—1661no—
307#41—1661no—
308#41—1661no—
309#41C22×D8311yescharacteristic
310#41C22×D8311yescharacteristic
311C2×Q8—831no—
312C2×Q8—1661no—
313C2×Q8—1661no—
314C2×D4—3321no—
315C2×D4—3321no—
316C2×D4—3321no—
317C2×D4—3321no—
318C2×D4—831no—
319C2×D4—831no—
320C2×D4—831no—
321C2×D4—1661no—
322C2×D4—1661no—
323C2×D4—1661no—
324C2×D4—1661no—
325C2×D4—1661no—
326C2×D4—1661no—
327Q16—831no—
328Q16—1661no—
329Q16—1661no—
330Q16D33211yescharacteristic, complemented
331SD16—831no—
332SD16—831no—
333SD16—1661no—
334SD16—1661no—
335SD16—1661no—
336SD16—1661no—
337SD16—1661no—
338SD16—1661no—
339SD16—1661no—
340SD16—1661no—
341SD16—1661no—
342SD16—1661no—
343SD16—1661no—
344SD16—1661no—
345SD16D33211yescharacteristic, complemented
346SD16D33211yescharacteristic, complemented
347D8—831no—
348D8—1661no—
349D8—1661no—
350D8—1661no—
351D8—1661no—
352D8—1661no—
353D8—1661no—
354D8—1661no—
355D8—1661no—
356D8—1661no—
357D8—1661no—
358D8D33211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—831no—
362M16—831no—
363M16—1661no—
364M16—1661no—
365M16C22×D8311yescharacteristic
366M16C22×D8311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—1661no—
370C2×C8—1661no—
371C2×C8C22×D8311yescharacteristic
372C2×C8C22×D8311yescharacteristic
373C22⋊C4—3321no—
374C42C22×D8311yescharacteristic
375C23—3321no—
376C23—3321no—
377C23—3321no—
378C23—1661no—
379C23—1661no—
380C23—1661no—
381Q8—3321no—
382Q8—831no—
383Q8—831no—
384Q8—1661no—
385Q8—1661no—
386Q8C2×D33211yescharacteristic
387Q8C2×D33211yescharacteristic
388D4—21no—
389D4—21no—
390D4—3321no—
391D4—3321no—
392D4—3321no—
393D4—3321no—
394D4—3321no—
395D4—3321no—
396D4—3321no—
397D4—831no—
398D4—831no—
399D4—6642no—
400D4—1661no—
401D4—1661no—
402D4—1661no—
403D4—1661no—
404D4—1661no—
405D4—1661no—
406D4—1661no—
407D4—1661no—
408D4C2×D33211yescharacteristic
409D4C2×D33211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—3321no—
414C2×C4—3321no—
415C2×C4—3321no—
416C2×C4—3321no—
417C2×C4—3321no—
418C2×C4—1661no—
419C2×C4—1661no—
420C2×C4—1661no—
421C2×C4C23×D8311yescharacteristic
422C8—21no—
423C8—21no—
424C8—1661no—
425C8—1661no—
426C8D4×D8311yescharacteristic
427C8D4×D8311yescharacteristic
428C8C2×D33211yescharacteristic
429C8C2×D33211yescharacteristic
430C22—21no—
431C22—21no—
432C22—13281no—
433C22—13281no—
434C22—13281no—
435C22—3321no—
436C22—3321no—
437C22—3321no—
438C22—6641no—
439C22—6641no—
440C22—6641no—
441C22—1661no—
442C22—1661no—
443C22#1458183811yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—3321no—
448C4—1661no—
449C4—1661no—
450C4C2×D4×D8311yescharacteristic
451C4C22×D33211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—3321no—
456C2—3321no—
457C2—6641no—
458C2—6641no—
459C2—6641no—
460C2D4×D33211yescharacteristic
subgroupG/H
centerC2D4×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#444×C83C2
socleC2×C83D42
hypercenter#444D83
hypocenterC83#3288
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #19128513C2semidirect
2. #19128508C2semidirect
3. #19128506C2semidirect
4. #19128503C2semidirect
5. #19128090C2semidirect
6. #19127972C2semidirect
7. #19127971C2semidirect
8. #19127965C2semidirect
9. #19127964C2semidirect
10. #19127880C2semidirect
11. #19127875C2semidirect
12. #19127870C2semidirect
13. #19127870C2semidirect
14. #19127824C2semidirect
15. #444×C83C2semidirect
16. #14581839C22semidirect
18. #14581838C22semidirect
19. #14581838C22semidirect
21. #14581825C22semidirect
23. #14581822C22semidirect
28. #14581820C22semidirect
29. #14581820C22semidirect
33. #14581817C22semidirect
34. #14581817C22semidirect
42. C2×#13050579C22semidirect
43. C2×#13050579C22semidirect
46. C2×D664C22semidirect
47. C2×D664C22semidirect
49. #14581798C22semidirect
50. #14581798C22semidirect
51. #14581789C22semidirect
52. #14581780C22semidirect
53. #14581728C22semidirect
54. #14581720C22semidirect
55. #14581720C22semidirect
56. #14581710C22semidirect
57. #14581708C22semidirect
58. #14581708C22semidirect
59. #14581708C22semidirect
61. #131×C83C22semidirect
62. #131×C83C22semidirect
63. #108×C83C22semidirect
64. #104×C83C22semidirect
65. #104×C83C22semidirect
66. #97×C83C22semidirect
95. Dic332D4semidirect
105. #13050579D4semidirect
106. #13050579D4semidirect
118. D664D4semidirect
131. Q16×C83D4semidirect
132. SD16×C83D4semidirect
133. SD16×C83D4semidirect
134. D8×C83D4semidirect
231. C83#3288semidirect
246. #444D83semidirect
283. #131C2×D83semidirect
284. #131C2×D83semidirect
288. #108C2×D83semidirect
291. #104C2×D83semidirect
292. #104C2×D83semidirect
297. #97C2×D83semidirect
330. Q16D332semidirect
345. SD16D332semidirect
346. SD16D332semidirect
358. D8D332semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#191285132
#191285082
#191285062
#191285032
#191280902
#191279722
#191279712
#191279652
#191279642
#191278802
#191278752
#191278702
#191278242
#444×C832
#328883